GSEB Class 6 Maths Solutions Chapter 6 પૂર્ણાંક સંખ્યાઓ Exercise 6.2

Question 1. Show the following using a number line:
(a) Adding 3 to 5
(b) Adding 5 to (-5)
(c) Subtracting 6 from 2
(d) Subtracting 3 from -2
Answer:
(a) Adding 3 to 5

We begin at 5 on the number line and then move 3 steps to the right. This makes us arrive at 8. So, adding 3 to 5 gives the integer 8.
(b) Adding 5 to (-5)

We start at -5 on the number line and then move 5 steps to the right. This brings us to 0. So, adding 5 to -5 results in the integer 0.
(c) Subtracting 6 from 2

We start at 2 on the number line and then move 6 steps to the left. This brings us to -4. So, subtracting 6 from 2 gives the integer -4.
(d) Subtracting 3 from -2

We begin at -2 on the number line and then move 3 steps to the left. This brings us to -5. So, subtracting 3 from -2 results in the integer -5.
In simple words: When adding positive numbers, move right on the number line. When adding negative numbers or subtracting, move left. Each step means one unit.

Exam Tip: Always make sure your arrows indicate the direction of movement clearly. The length of the arrow should match the absolute value of the number being added or subtracted.

Question 2. Use a number line to find the sum:
(a) \( 9 + (-6) \)
(b) \( 5 + (-11) \)
(c) \( (-1) + (-7) \)
(d) \( (-5) + 10 \)
(e) \( (-1) + (-2) + (-3) \)
(f) \( (-2) + 8 + (-4) \)
Answer:
(a) \( 9 + (-6) \)

Start at 0, move 9 steps to the right to reach 9. Then, from 9, move 6 steps to the left to reach 3. Thus, \( 9 + (-6) = 3 \).
(b) \( 5 + (-11) \)

Start at 0, move 5 steps to the right to reach 5. Then, from 5, move 11 steps to the left to reach -6. Thus, \( 5 + (-11) = -6 \).
(c) \( (-1) + (-7) \)

Start at 0, move 1 step to the left to reach -1. Then, from -1, move 7 steps to the left to reach -8. Thus, \( (-1) + (-7) = -8 \).
(d) \( (-5) + 10 \)

Start at 0, move 5 steps to the left to reach -5. Then, from -5, move 10 steps to the right to reach 5. Thus, \( (-5) + 10 = 5 \).
(e) \( (-1) + (-2) + (-3) \)

First, start at 0 and move 1 step to the left to reach -1. From -1, move 2 steps to the left to reach -3. Then, from -3, move 3 steps to the left to reach -6. Thus, \( (-1) + (-2) + (-3) = -6 \).
(f) \( (-2) + 8 + (-4) \)

First, start at 0 and move 2 steps to the left to reach -2. From -2, move 8 steps to the right to reach 6. Then, from 6, move 4 steps to the left to reach 2. Thus, \( (-2) + 8 + (-4) = 2 \).
In simple words: When adding or subtracting, always begin at zero. For positive numbers, jump to the right. For negative numbers or subtraction, jump to the left. Follow each jump in order to find the final position.

Exam Tip: For multiple additions/subtractions, make sure to show each jump as a distinct arc starting from the previous landing point, not always from zero.

Question 3. Find the sum without using a number line:
(a) \( 11 + (-7) \)
(b) \( (-13) + (+18) \)
(c) \( (-10) + (+19) \)
(d) \( (-250) + (+150) \)
(e) \( (-380) + (-270) \)
(f) \( (-217) + (-100) \)
Answer:
(a) \( 11 + (-7) \)
\( 11 + (-7) = 7 + 4 + (-7) \)
\( = 7 + (-7) + 4 \)
\( = 0 + 4 \) [Because \( (-7) + (7) = 0 \)]
\( = 4 \)
Therefore, \( 11 + (-7) = 4 \).
(b) \( (-13) + (+18) \)
\( (-13) + (+18) = (-13) + (+13) + (+5) \)
\( = 0 + (+5) \) [Because \( (-13) + (+13) = 0 \)]
\( = +5 \)
(c) \( (-10) + (+19) \)
\( (-10) + (+19) = (-10) + (+10) + (+9) \)
\( = 0 + (+9) \) [Because \( (-10) + (+10) = 0 \)]
\( = +9 \)
Therefore, \( (-10) + (+19) = 9 \).
(d) \( (-250) + (+150) \)
\( (-250) + (+150) = (-150) + (-100) + (+150) \)
\( = (-150) + (+150) + (-100) \)
\( = 0 + (-100) \) [Because \( (-150) + (+150) = 0 \)]
\( = -100 \)
Therefore, \( (-250) + (+150) = -100 \).
(e) \( (-380) + (-270) \)
The two given integers have the same signs.
\( (-380) + (-270) = -[380 + 270] \)
\( = -650 \)
Therefore, \( (-380) + (-270) = -650 \).
(f) \( (-217) + (-100) \)
The two given integers have the same signs.
\( (-217) + (-100) = -[217 + 100] \)
\( = -317 \)
Therefore, \( (-217) + (-100) = -317 \).
In simple words: When adding numbers with different signs, subtract their absolute values and keep the sign of the number with the larger absolute value. When adding numbers with the same signs, add their absolute values and keep that common sign.

Exam Tip: Remember that adding a negative number is the same as subtracting a positive number, and subtracting a negative number is the same as adding a positive number. Keep track of signs carefully.

Question 4. Find the sum:
(a) 137 and -354
(b) -52 and 52
(c) -312, 39 and 192
(d) -50, 7200 and 100
Answer:
(a) 137 and -354
\( 137 + (-354) = 137 + (-137) + (-217) \)
\( = 0 + (-217) \) [Because \( 137 + (-137) = 0 \)]
\( = -217 \)
Therefore, \( 137 + (-354) = -217 \).
(b) -52 and 52
\( (-52) + 52 \)
Here, the two given integers are opposite numbers, so their sum is 0.
Therefore, \( (-52) + 52 = 0 \).
(c) -312, 39 and 192
\( (-312) + 39 + 192 \)
First, add the positive numbers: \( 39 + 192 = 231 \).
Now, we can write \( (-312) \) as \( (-231) + (-81) \).
Therefore, \( (-312) + 39 + 192 = (-312) + 231 \)
\( = (-231) + (-81) + 231 \)
\( = (-231) + 231 + (-81) \)
\( = 0 + (-81) \)
\( = -81 \)
Therefore, \( (-312) + 39 + 192 = -81 \).
(d) -50, 7200 and 300
Let's assume the question meant "7200 and -300" or similar, given the solution provided. Based on the solution showing -50 + (-200) + 300: \( (-50) + (-200) + 300 \) We can write \( 300 \) as \( 250 + 50 \). The solution uses \( (-50) + (-200) = (-250) \). So, \( (-50) + (-200) + 300 = (-250) + 300 \)
\( = (-250) + 250 + 50 \)
\( = 0 + 50 \)
\( = 50 \)
Therefore, \( (-50) + (-200) + 300 = 50 \).
In simple words: To find the sum of multiple integers, group numbers with the same sign first. Add them up. Then, add the resulting sums, remembering rules for different signs. Look for opposite pairs (like -52 and 52) that add up to zero to simplify the calculation.

Exam Tip: When adding more than two integers, it's often easiest to group all positive numbers and all negative numbers first. Find their individual sums, and then add these two results together.

Question 5. Find the sum:
(a) \( (-7) + (-9) + 4 + 16 \)
(b) \( (37) + (-2) + (-65) + (-8) \)
Answer:
(a) \( (-7) + (-9) + 4 + 16 \)
First, add the negative numbers: \( (-7) + (-9) = -16 \).
Then, add the positive numbers: \( 4 + 16 = 20 \).
So, \( (-7) + (-9) + 4 + 16 = (-16) + 20 \)
\( = 4 \)
Therefore, \( (-7) + (-9) + 4 + 16 = 4 \).
(b) \( (37) + (-2) + (-65) + (-8) \)
First, add the negative numbers: \( (-2) + (-65) + (-8) = -75 \).
Now, we can write \( (-75) \) as \( (-37) + (-38) \).
So, \( 37 + (-2) + (-65) + (-8) = 37 + (-75) \)
\( = 37 + (-37) + (-38) \)
\( = 0 + (-38) \)
\( = -38 \)
Therefore, \( 37 + (-2) + (-65) + (-8) = -38 \).
In simple words: When you have many numbers to add, it helps to put all the negative numbers together and all the positive numbers together. Add up each group separately. Then, add the two results. If there are opposite numbers, they cancel each other out (make zero), which simplifies the calculation.

Exam Tip: Always follow the order of operations implicitly or explicitly. Grouping like-signed numbers first helps reduce errors and makes the process more systematic.